# Shock cords

[I’ve written about shock cords before. But I have more information now, and have done some more thinking, so I want to take a fresh attempt at writing some of it up.]

A typical hobby rocket separates into two sections at or near apogee — call them the airframe and the nose — which are connected to one another, and often to a parachute or streamer, by a shock cord. One thing that’s fairly obvious, I hope, is that a shock cord’s not doing its job if it breaks in two! How do you minimize the chances of that happening? It’s not easy to work this out with any kind of rigor. But you can at least get a feeling for some of the issues with a simplified model.

A number of materials have been used for shock cords, but two of them would be rubber (the standard Estes rubber bands, for instance) and Kevlar. A typical Kevlar shock cord, of course, requires much more force to break than a typical rubber one does, so does that make Kevlar the better choice? Of course not; there are other considerations, like cost for instance. But even just considering the physics, the foregoing reasoning isn’t right.

How much force does a shock cord need to withstand, anyway? It depends on the rocket, of course, and it’s not just “the bigger the rocket, the bigger the force”. In fact, if you compare say a Baby Bertha and a Big Bertha, the former has a much smaller volume for the ejection gases to pressurize, which suggests it’ll eject its nose cone more vigorously; the Baby may need a stronger shock cord than the Big! It depends, too, on the size of the ejection charge. All that’s obvious, but what might not be obvious is that the force on the shock cord depends on the shock cord.

Real world rockets and shock cords have too complex behavior for easy analysis, so let’s just simplify the problem. We won’t get accurate real world numbers out of it but we’ll see what’s going on. I’ll assume the shock cord is a perfect, massless rubber band. By “perfect” I mean it obeys Hooke’s Law under all circumstances. That is, the force $F$ it takes to stretch it a distance $x$ is proportional to $x$: $F = kx$$k$ is a proportionality constant having to do with the length of the shock cord (longer cord, lower $k$), the cross sectional area of the shock cord (larger area, larger $k$), and the material it’s made of. The shock cord’s massless because if it droops under its own weight, that complicates things greatly. I’m also neglecting the effects of wind and air resistance, and ignoring the existence of a parachute. In terms familiar to freshman physics students, we’re modeling the system as two masses connected by an ideal spring.

What’s happening in a rubber band is you have crumpled-up polymer molecules which, as you stretch the band, straighten out. It takes energy to straighten out a molecule. So when you’re stretching a rubber band, you’re storing energy in it. In a real rubber band any given molecule can only store so much energy before it straightens out as far as it can, and there are only so many molecules present, so there’s a limit to how much energy you can store in the band before you start pulling molecular bonds apart, damaging and ultimately breaking the rubber. But for our ideal rubber band we’ll assume that never happens, at least not at the distances we’ll be stretching it.

When a nose cone and airframe go flying in opposite directions, they stretch the shock cord, storing energy in it. That energy has to come from somewhere: from the kinetic energy of the rocket components. Initially the nose cone has a velocity $v_c$ so its kinetic energy is $E_c = \frac{1}{2}m_cv_c^2$, where $m_c$ is the nose cone mass. Likewise the kinetic energy of the airframe is $E_a = \frac{1}{2}m_av_a^2$ where $m_a$ and $v_a$ are the mass and velocity of the airframe. The sum of the two is $E = E_c + E_a$, the total kinetic energy of the rocket components. The shock cord absorbs all that energy in the act of stopping the nose cone and airframe from flying apart.

Now, if the force $F = kx$, you can integrate that to get the energy stored in a stretched rubber band: $E = \frac{1}{2}kx^2$Solving that for $x$ gives you $x = \sqrt {\frac{2E}{k}}$, and that means the maximum force on the shock cord (at the instant it’s stretched to the maximum) is $F = \sqrt {2kE}$. The force depends on $E$, which depends on the properties of the rocket and the ejection charge, but it also depends on $k$, a property of the shock cord! A stretchier shock cord (smaller $k$) experiences a smaller maximum force than a stiffer, harder-to-stretch one.

What about Kevlar? You might not think of Kevlar as stretchy, but it does stretch when you pull on it. It takes a lot more force to stretch Kevlar by a given amount than for rubber, and you can’t stretch it very far before it deforms and breaks. But if you don’t stretch it that far, you can think of it as a very, very stiff piece of elastic; one with a much higher value of $k$ than an Estes rubber band. What that means is, if you replace a rubber band shock cord with a piece of Kevlar, it’ll experience much larger forces than the rubber band would have.

Of course, the good news is, Kevlar can withstand much larger forces than rubber. Though really we should be talking about energy, not force: if you subject a rubber band to a large enough constant force it’ll break, but it might withstand that force for a short period of time, and in a rocket shock cord the force isn’t constant. But as I said above, a real world shock cord can absorb only so much energy before it deforms and breaks — whether you stretch it slowly or abruptly. For a typical Kevlar shock cord, that maximum energy capacity is a lot larger than for an Estes rubber band. So the good news is, if you think correctly in terms of energy, Kevlar will be less likely to break. The bad news is, it’ll experience much higher force.

Why is that bad news, if the Kevlar can withstand the strain? Well, remember Isaac Newton? Equal and opposite reaction? If the nose cone and airframe are exerting a much higher force on a Kevlar shock cord, then the Kevlar shock cord is exerting a much higher force on the nose cone and airframe. The Kevlar can take it, but can the shock cord anchors — the loop on the nose cone and the trifold mount on the airframe? Well, of course you’re likely to have ditched the trifold in favor of tying the Kevlar to the motor mount, but what about that nose cone loop?

Wait a second, though. If force isn’t really the way to think about what’ll break a shock cord, it’s also not really the way to think about what’ll break a shock cord anchor. If you don’t think of Kevlar as stretchy then that probably goes double for a nose cone loop, but again, like basically any solid material, the plastic will stretch slightly and absorb a certain amount of energy before it deforms and breaks. So as an abstract model, we can think of the rocket as two masses connected by three springs in series, end to end. Springs 1 and 3 represent the shock cord anchors while spring 2 is the shock cord itself, and, if we treat all three as ideal springs, they each have their own $k$ value: $k_1$$k_2$, and $k_3$. Noting that the force on all three springs must be the same we can work out that $E = \frac{1}{2}kx^2$ where $k = 1/(1/k_1+1/k_2+1/k_3)$. Stare at this a little and you’ll realize that if the shock cord is a rubber band and the anchors are glue, cardboard, and plastic, $k_2$ will be much smaller than $k_1$ and $k_3$, so $1/k_2$ will be much larger than $1/k_1$ and $1/k_3$, which means to a good approximation, $k = k_2$. Furthermore the fraction of the energy absorbed by the anchors will be $k/k_1$ and $k/k_3$ — very small fractions if, again, the shock cord has much lower $k$ than the anchors. But if you double the value of $k$, you double the energy absorbed by the anchors. The approximation breaks down as $k_2$ approaches $k_1$ and $k_3$ but the basic idea is still true: if you replace a rubber band shock cord with Kevlar, then the share of the energy the shock cord anchors have to absorb increases a lot. In other words, the likelihood of the shock cord failing is lower… but the likelihood of the anchors failing is higher.

There are things you can do to mitigate that, of course. Like change the mounts. As I’ve said, you probably don’t want to use a trifold mount with Kevlar in a big rocket. You also can find a beefier way to attach to the nose cone. But in addition, you can reduce $k_2$: make the shock cord longer. A 180 cm shock cord will lower the energy load on the anchors by a factor of 3 compared to a 60 cm shock cord. (Just don’t blindly follow the “rule of thumb” that a shock cord should be so many times the length of the rocket. From an anchor damage standpoint, at least — and granted there are other considerations in choosing a shock cord length — there’s no logic behind that “rule” and in some cases, like the Baby vs. Big Bertha, it gets it exactly backwards.)

Or just use an elastic shock cord: For most low and mid power rockets, elastic in good condition should be easily capable of absorbing the energy. A Kevlar leader lets you anchor to the motor mount, avoiding the drawbacks of a trifold, while keeping the elastic away from the heat, and enables you to change out the elastic easily when it wears out or becomes damaged. If the leader ends behind the forward end of the body tube it won’t cause a zipper. The lower $k$ of the elastic means it will absorb most of the energy, protecting the shock cord anchors. For high power it’s a different story but up to there, elastic with a Kevlar leader has a lot to recommend it.